Bounding the Space of Holographic CFTs with Chaos

Abstract

Thermal states of quantum systems with many degrees of freedom are subject to a bound on the rate of onset of chaos, including a bound on the Lyapunov exponent, ${\lambda }_L\le 2\pi /\beta$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying out-of-time-order four-point functions, we discuss how ${\lambda }_L=2\pi /\beta$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large central charge and a set of higher spin currents of bounded spin; this implies the inconsistency of weakly coupled AdS${}_3$ higher spin gravities without infinite towers of gauge fields, such as the $SL(N)$ theories. This fits naturally with the structure of higher-dimensional gravity, where finite towers of higher spin fields lead to acausality. On the other hand, unitary CFTs with classical $W_{\infty }[\lambda ]$ symmetry, dual to 3D Vasiliev or hs[$\lambda$] higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: ${\lambda }_L=0$. Independently, we show that such theories violate unitarity for $|\lambda |>2$. These results encourage a tensionless string theory interpretation of the 3D Vasiliev theory. We also perform some CFT calculations of chaos in Rindler space in various dimensions.

50+18 pages. v3: Section 3 clarified and reorganized, related improvements

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